## Abstract

We consider the invariant measure of homogeneous random walks in the quarter-plane. In particular, we consider measures that can be expressed as a countably infinite sum of geometric terms which individually satisfy the interior balance equations. We demonstrate that the compensation approach is the only method that may lead to such a type of invariant measure. In particular, we show that if a countably infinite sum of geometric terms is an invariant measure, then the geometric terms in an invariant measure must be the union of at most six pairwise-coupled sets of countably infinite cardinality each. We further show that for such invariant measure to be a countably infinite sum of geometric terms, the random walk cannot have transitions to the north, northeast or east. Finally, we show that for a countably infinite weighted sum of geometric terms to be an invariant measure at least one of the weights must be negative.

Original language | English |
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Pages (from-to) | 257-277 |

Number of pages | 20 |

Journal | Queueing systems |

Volume | 94 |

Issue number | 3-4 |

Early online date | 8 Jul 2019 |

DOIs | |

Publication status | Published - 1 Apr 2020 |

## Keywords

- UT-Hybrid-D
- Compensation approach
- Geometric term
- Invariant measure
- Pairwise-coupled
- Quarter-plane
- Random walk
- Algebraic curve